Deck 15: Applications of the Schrodinger Equation

Question 1

For the finite square well shown in the figure,

A) U(x) = U₀, x < 0
B) U(x) = 0, 0 < x < L
C) U(x) = U₀, x > 0
D) U, although discontinuous, is finite everywhere.
E) All of these are correct.

Answer 1

All of these are correct.

Question 2

The graph that shows the third state for a particle in a finite square well is

A) 1
B) 2
C) 3
D) 4
E) None of these is correct.

Answer 2

3

Question 3

The graph that represents the potential energy for a one-dimensional box from x = 0 to x = L is

A) 1
B) 2
C) 3
D) 4
E) 5

Answer 3

4

Question 4

The solution for the time-independent Schrödinger equation with the following potential function and boundary conditions
U(x) = 0, -L/2 $\le$ x $\le$ L/2
U(x) = $\infty$ , x < -L/2 or x > L/2
Is

A) , where and n = 1, 2, 3, ..
B) , where and n = 1, 2, 3, ..
C) , where and n = 1, 2, 3, ..
D) , where and n = 1, 2, 3, ..
E) There is no solution for the given potential function and boundary conditions.

Question 5

Suppose the natural frequency of oscillation for H₂ is $\backsim$ 10¹² Hz, the effective spring constant between the two H atoms is of the order

A) 10^-1 N/m
B) 10^-3 N/m
C) 10^-5 N/m
D) 10^-7 N/m
E) 10^-9 N/m

Question 6

The curve that best represents the ground-state energy for a finite square well is

A) 1
B) 2
C) 3
D) 4
E) 5

Question 7

The energy in the first excited state of an electron confined to a one-dimensional box of length L = 0.2 nm is

A) 9.40 eV
B) 12.3 eV
C) 19.7 eV
D) 24.2 eV
E) 37.6 eV

Question 8

The energy in the ground state of an electron confined to a one-dimensional box of length L = 0.3 nm is

A) 2.47 eV
B) 4.18 eV
C) 6.25 eV
D) 9.40 eV
E) None of these is correct.

Question 9

The energy in the ground state of an electron confined to a one-dimensional box of length L = 0.2 nm is

A) 1.88 eV
B) 4.47 eV
C) 6.25 eV
D) 9.40 eV
E) None of these is correct.

Question 10

Match each of the wave function on the left for a particle in a finite square well with the corresponding probability distributions on the right, in the order (a), (b), and (c).
(a) (I) (b) (II) (c) (III)

A) (II) (I) (III)
B) (II) (III) (I)
C) (III) (I) (II)
D) (III) (II) (I)
E) (I) (III) (II)

Question 11

The penetration of the wave function beyond the edges of the finite square-well potential shown in the figure

A) indicates that there is some small probability of finding the particle outside the well.
B) forces us to conclude that E - U₀ is negative in these regions.
C) suggests that these regions are classically forbidden.
D) is unexplainable classically.
E) is described by all of the above.

Question 12

An electron confined to a one-dimensional box of length L = 0.2 nm makes a transition from state n = 4 to state n = 3. The wavelength of the photon emitted is

A) 19.0 nm
B) 17.2 nm
C) 14.6 nm
D) 12.5 nm
E) 10.8 nm

Question 13

The energy in the first excited state of an electron confined to a one-dimensional box of length L = 0.3 nm is

A) 9.40 eV
B) 12.3 eV
C) 16.7 eV
D) 24.2 eV
E) 37.6 eV

Question 14

The dependent variable in the Schrödinger equation is

A) the wave function $\Psi$ .
B) the position variable x.
C) the time variable t.
D) the potential energy function U.
E) None of these is correct.

Question 15

The wave function for a particle in a one-dimensional box of length L

A) is constrained by the boundary conditions $\Psi$ (0) = 0 and $\Psi$ (L) = 0.
B) must be zero everywhere outside of the box.
C) is given by $\Psi$ (x) = A sin kx, where A is a constant.
D) restricts the possible energy of the particle to E = .
E) All of these are correct.

Question 16

An electron confined to a one-dimensional box of length L = 0.3 nm makes a transition from state n = 4 to state n = 2. The wavelength of the photon emitted is

A) 19.0 nm
B) 24.7 nm
C) 28.9 nm
D) 33.6 nm
E) 41.2 nm

Question 17

The Schrödinger equation

A) is a partial differential equation in space and time.
B) (like Newton's laws of motion) cannot be derived.
C) depends upon experimentation for its verification.
D) relates the second space-derivative of the wave function to the first time-derivative of the wave function.
E) All of these are correct.

Question 18

In order to solve the Schrödinger's equation, which of the following quantity(ies) must be specified?

A) the kinetic energy function of the particle
B) the potential energy function of the particle
C) the boundary conditions
D) (A) and (B)
E) (B) and (C)

Question 19

The time-independent Schrödinger equation is

A)
B)
C)
D)
E)

Question 20

The graph that shows the second state for a particle in a finite square well is

A) 1
B) 2
C) 3
D) 4
E) None of these is correct.

Question 21

When a particle of energy E encounters the step function shown in the figure,

A) the classical and quantum-mechanical descriptions are in agreement provided
E < U₀.
B) the wave function does not go to zero at x = 0 but rather decays exponentially.
C) the particle will sometimes be transmitted and sometimes reflected if E > U₀.
D) the wavelength of the particle changes abruptly at x = 0 if E > U₀.
E) All of these are true.

Question 22

The ground-state wave function of the harmonic oscillator is best represented by

A) 1
B) 2
C) 3
D) 4
E) None of these is correct.

Question 23

The ground-state wave function of the harmonic oscillator is

A)
B) $\Psi$ ₀(x) = A₀e^-ax
C) $\Psi$ ₀(x) = A₀sin(ax)
D) $\Psi$ ₀(x) = A₀sin(ax²)
E)

Question 24

A proton of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀. The probability that the proton will be transmitted is

A) 85.3%
B) 89.2%
C) 92.4%
D) 97.1%
E) 98.3%

Question 25

A particle of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.3E₀. The probability that the particle will be reflected is

A) 0.316%
B) 0.791%
C) 2.89%
D) 3.56%
E) 4.12%

Question 26

The wave function shown in the figure represents the n = _______ energy state of the harmonic oscillator.

A) 0
B) 1
C) 2
D) 3
E) 4

Question 27

The probability of penetration of a rectangular barrier __________ with the square root of the relative barrier height.

A) increases linearly
B) decreases exponentially
C) decreases linearly
D) increases exponentially
E) decreases as a damped sinusoid

Question 28

The probability of penetration of a rectangular barrier decreases exponentially with the ________ of the barrier height.

A) square
B) cube
C) square root
D) cube root
E) fourth root

Question 29

Suppose the natural frequency of oscillation for H₂ is $\backsim$ 10¹² Hz and the amplitude of oscillation is $\backsim$ 10^-12 m, the total energy of the harmonic oscillator is of the order

A) 10^-20 J
B) 10^-22 J
C) 10^-24 J
D) 10^-26 J
E) 10^-28 J

Question 30

The probability of penetration of a rectangular barrier ________ with the barrier thickness.

A) increases linearly
B) decreases exponentially
C) decreases linearly
D) increases exponentially
E) decreases as a damped sinusoid

Question 31

The potential energy function shown in the above figure

A) applies to any system undergoing small oscillations about a position of stable equilibrium.
B) shows the classical turning points ±A.
C) tells that where -A < x < +A, the total energy is greater than the potential energy.
D) indicates that U(x) is directly proportional to x².
E) All of these are correct.

Question 32

The wave function shown in the figure represents the n = _______ energy state of the harmonic oscillator.

A) 0
B) 1
C) 2
D) 3
E) 4

Question 33

Which of the following statements is true?

A) A particle that is confined to some region of space can have zero energy.
B) All phenomena in nature are adequately described by classical wave theory.
C) The Schrödinger equation can be derived from Newton's laws of motion.
D) The penetration of a barrier by a wave has physical significance.
E) None of these is true.

Question 34

The quantum phenomenon known as the "tunnel effect" refers to

A) highly eccentric electron orbits penetrating inner closed shells.
B) the fine structure exhibited by many spectral lines.
C) the small but finite probability that an $\alpha$ -particle originally within the nucleus will be found outside the nucleus.
D) the penetration of shielding by high-energy fission neutrons.
E) an orbital electron penetrating the nucleus and undergoing electron capture.

Question 35

A particle of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U₀. What is the ratio of E₀/U₀ so that the reflection co-efficient is 75%? (Assume E₀ is much less than the rest mass energy of the particle.)

A) 1.250
B) 0.7500
C) 1.778
D) 1.005
E) 1.063

Question 36

A particle of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U₀. What is the ratio of E₀/U₀ so that the reflection co-efficient is 25%? (Assume E₀ is much less than the rest mass energy of the particle.)

A) 4.0
B) 1.25
C) 1.125
D) 1.025
E) 0.75

Question 37

An electron of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀. The ratio of the wavelength of the transmitted wave to the incident wave is

A) 2
B) 1
C) 0.5
D)
E)

Question 38

The wave function shown in the figure represents the n = _______ energy state of the harmonic oscillator.

A) 0
B) 1
C) 2
D) 3
E) 4

Question 39

A particle of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.4E₀. The probability that the particle will be reflected is

A) 0.316%
B) 0.789%
C) 1.61%
D) 3.56%
E) 4.12%

Question 40

An electron with kinetic energy 5.0 eV traveling in a region in which the potential energy is zero is incident at x > 0 on a potential barrier of height 3.0 eV. What is the wavelength of the electron in the region x > 0?

A) 0.87 nm
B) 0.99 nm
C) 0.54 nm
D) 2.3 nm
E) 0.14 nm

Question 41

An electron is confined in a two-dimensional box where U(x,y) = 0 for x = 0 to L, and y = 0 to 3L, and U(x,y) = infinity outside these boundaries. If L = 0.5 nm then calculate the energy of the first doubly degenerate levels.

A) 9.9 eV
B) 11.0 eV
C) 8.5 eV
D) 7.7 eV
E) 6.2 eV

Question 42

A particle is in a three-dimensional box with L₃ = L₂ = 3L₁. The lowest energy level is

A) zero
B) 1.22E₀
C) 1.56E₀
D) 1.67E₀
E) 1.94E₀

Question 43

An electron is confined in a two-dimensional box where U(x,y) = 0 for x = 0 to L and y = 0 to 3L, and U(x,y) = infinity outside these boundaries. If L = 0.5 nm, then calculate the energy of the first excited state.

A) 1.7 eV
B) 2.2 eV
C) 6.2 eV
D) 3.0 eV
E) None of these is correct.

Question 44

Particles that have antisymmetric wave functions and are described by the Pauli exclusion principle are called

A) quarks
B) leptons
C) fermions
D) bosons
E) None of these is correct.

Question 45

A particle is confined in a three-dimensional box with L₁ = L₂ = 3L₃. The quantum numbers for the second excited state are

A) (1,1,2) and (1,2,1)
B) (1,2,1) and (2,1,1)
C) (2,2,1) and (2,1,2)
D) (1,2,2) and (2,1,2)
E) (2,2,1) and (1,2,2)

Question 46

A particle is in a three-dimensional box with L₃ = L₂ = 3L₁. The energy level E_1,1,2 is

A) zero
B) 1.22E₀
C) 1.56E₀
D) 1.67E₀
E) 1.94E₀

Question 47

An electron of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a finite potential barrier of height U₀ (= 4E₀) and width a. If the potential barrier is reduced to 2E₀, by what factor will the probability of penetration of the barrier be changed?

A)
B)
C)
D)
E) None of these is correct.

Question 48

The wave function for the energy level in a cubical box of side L that corresponds to the quantum numbers 1, 2, and 3 is

A) $\Psi$ _1,2,3 = A sin( $\pi$ x/L) sin(2 $\pi$ x/L) sin( $\pi$ x/L)
B) $\Psi$ _1,2,3 = A sin( $\pi$ x/L) sin( $\pi$ x/L) sin(3 $\pi$ x/L)
C) $\Psi$ _1,2,3 = sin(2 $\pi$ x/L) sin(3 $\pi$ x/L)
D) $\Psi$ _1,2,3 = A sin( $\pi$ x/L) sin(2 $\pi$ x/L) sin(3 $\pi$ x/L)
E) $\Psi$ _1,2,3 = 2A sin( $\pi$ x/L) sin(2 $\pi$ x/L) sin(3 $\pi$ x/L)

Question 49

In three dimensions, the Schrödinger equation for the infinite square-well potential

A) has a solution of the form $\Psi$ (x,y,z) = A sin k₁x sin k₂y sin k₃z, where the k's are wave numbers and the constant A is determined by normalization.
B) predicts energy states described by .
C) predicts energies and wave functions that are characterized by three quantum numbers.
D) allows multiple quantum states corresponding to the same energy level.
E) All of these are true.

Question 50

A particle is confined in a three-dimensional box with L₁ = L, L₂ = 2L and L₃ = 3L. The energy levels of the particle are given by

A) .
B) .
C) .
D) .
E) None of these is correct.

Question 51

A particle of mass m is confined in a two-dimensional box that has sides L_x = L and L_y = 2L. By what factor is the energy of the 3rd excited state larger than the energy of the ground state?

A) 5/4
B) 13/5
C) 17/4
D) 17/5
E) 4

Question 52

You put 5 non-interacting identical fermions each of mass m into a 1-d box of dimension L. You then put 10 non-interacting bosons each of mass m into a 1-d box of length 2L. Which system has the lowest ground-state energy and what is the value of the fermion system ground-state energy divided by the boson system ground-state energy?

A) fermion system, 19/10
B) boson system, 10/19
C) boson system, 38/5
D) fermion system, 5/19
E) none of the above

Question 53

Particles that have symmetric wave functions and are not subject to the Pauli exclusion principle are called

A) quarks
B) leptons
C) fermions
D) bosons
E) None of these is correct.

Question 54

The number of degenerate states in the third excited state for a particle in a three-dimensional box with L₁ = L₂ = L₃ is

A) 2
B) 3
C) 4
D) 5
E) 6